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Abstracts for Conference on "Arithmetic Geometry", July 22 - 26, 2024

Alternatively have a look at the program.

Heights and periods of homologically trivial cycles in families, I (live stream)

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Speaker: 
Shou-Wu Zhang
Affiliation: 
Princeton University
Date: 
Mon, 22/07/2024 - 09:30 - 10:30
Location: 
MPIM Lecture Hall

In these two talks, we consider the Beilinson—Bloch heights and Abel—Jacobian periods  of homologically trivial Chow cycles in families. For the Beilinson-Bloch  heights, we show that for any  $g>2$, there is a non-empty Zariski open subset $U$ of $M_g$, the coarse moduli of curves of genus $g$ over rationals, such that the heights of Ceresa cycles and Gross—Schoen cycles over $U$ satisfy the Northcott property.

Heights and periods of homologically trivial cycles in families, II (live stream)

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Speaker: 
Ziyang Gao
Affiliation: 
Leibniz Universität Hannover
Date: 
Mon, 22/07/2024 - 11:00 - 12:00
Location: 
MPIM Lecture Hall

In these two talks, we consider the Beilinson—Bloch heights and Abel—Jacobian periods  of homologically trivial Chow cycles in families. For the Beilinson-Bloch  heights, we show that for any  $g>2$, there is a non-empty Zariski open subset $U$ of $M_g$, the coarse moduli of curves of genus $g$ over rationals, such that the heights of Ceresa cycles and Gross—Schoen cycles over $U$ satisfy the Northcott property.

Calculus on formal-analytic arithmetic surfaces (live stream)

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Speaker: 
Jean-Benoit Bost
Affiliation: 
Université Paris-Saclay
Date: 
Mon, 22/07/2024 - 15:00 - 16:00
Location: 
MPIM Lecture Hall

Formal-analytic arithmetic surfaces are arithmetic analogues of germs of formal surfaces along a proper curve over some base field, in the same way as arithmetic surfaces are analogues of projective algebraic surfaces over some base field. It is possible to develop a  calculus  on formal-analytic arithmetic surfaces, involving Hermitian vector bundles and suitably defined arithmetic intersection numbers à la Arakelov. This calculus admits applications to finiteness results concerning the étale fundamental group of arithmetic surfaces, notably of integral models of modular curves.

A positive proportion of abelian surfaces are modular (live stream)

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Speaker: 
Toby Gee
Affiliation: 
Imperial College London
Date: 
Mon, 22/07/2024 - 16:30 - 17:30
Location: 
MPIM Lecture Hall

I will discuss joint work with George Boxer, Frank Calegari, and Vincent Pilloni in which we prove the modularity of a positive proportion of abelian surfaces over Q.

tba (live stream)

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Speaker: 
Michael Harris
Affiliation: 
Columbia University
Date: 
Tue, 23/07/2024 - 09:30 - 10:30
Location: 
MPIM Lecture Hall

tba (see website and notice board)

Faltings heights and the sub-leading terms of adjoint L-functions (live stream)

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Speaker: 
Wei Zhang
Affiliation: 
MIT
Date: 
Tue, 23/07/2024 - 11:00 - 12:00
Location: 
MPIM Lecture Hall

The Kronecker limit formula may be interpreted as an equality relating the Faltings height of an CM elliptic curve to the sub-leading term (at s=0) of the Dirichlet L-function of an imaginary quadratic character. Colmez conjectured a generalization relating the Faltings height of any CM abelian variety to the subheading terms of certain Artin L-functions. An averaged (over all abelian varieties with CM by the integer ring of a given CM field) version was proved by Andreatta—Goren—Howard—Madapusi and Yuan—Zhang.

On an Eichler-Shimura decomposition for ordinary p-adic Siegel modular forms (live stream)

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Speaker: 
Ana Caraiani
Affiliation: 
Imperial College London
Date: 
Tue, 23/07/2024 - 15:00 - 16:00
Location: 
MPIM Lecture Hall

There are two different ways to construct families of ordinary p-adic Siegel modular forms. One is by p-adically interpolating classes in Betti cohomology, first introduced by Hida and then given a more representation-theoretic interpretation by Emerton. The other is by p-adically interpolating classes in coherent cohomology, once again pioneered by Hida and generalised in recent years by Boxer and Pilloni. I will explain these two constructions and then discuss joint work in progress with James Newton and Juan Esteban Rodríguez Camargo that aims to compare them.

Calculus on moduli spaces of curves (live stream)

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Speaker: 
Robin de Jong
Affiliation: 
Universiteit Leiden
Date: 
Tue, 23/07/2024 - 16:30 - 17:30
Location: 
MPIM Lecture Hall

The purpose of this talk is to survey Gerd Faltings’s contributions to the Arakelov geometry of curves. The groundwork was laid in his paper “Calculus on arithmetic surfaces”, finished in early 1983, and published in Annals in 1984. As Faltings writes in the introduction, the results of this paper turned out to be “extremely useful” to the author in finding his proof, in May 1983, of the Mordell conjecture. We present some of the fundamental ideas from loc. cit., and then proceed to discuss recent developments related to its contents.

Tame $\mathcal G$-bundles on curves (live stream)

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Speaker: 
Michael Rapoport
Affiliation: 
University of Bonn
Date: 
Wed, 24/07/2024 - 09:30 - 10:30
Location: 
MPIM Lecture Hall

I will describe the objects one obtains by descent from $G$-bundles on a tame Galois cover of a given curve, answering a question raised by Grothendieck in his Bourbaki talk in 1956 on Weil's "memoire paper" from 1938. Joint work with George Pappas.

A pro-algebraic fundamental group for topological spaces and an arithmetic application (live stream)

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Speaker: 
Christopher Deninger
Affiliation: 
Universität Münster
Date: 
Wed, 24/07/2024 - 11:00 - 12:00
Location: 
MPIM Lecture Hall

Using the Tannakian formalism we defined and studied a pro-algebraic fundamental group for connected topological spaces. Using ideas of Nori and a result of Deligne on fibre functors of Tannakian categories we also defined a pseudo-torsor under this fundamental group which can serve as a replacement for the universal covering space in this generality. We introduce amalgamated products of pro-algebraic groups in order to prove a Seifert van Kampen theorem.

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